有界区域上非线性 Schrodinger 方程解的全局性质

本文解决了 H.Bréis 在文[1]中提出的一个问题,对于非线性 Schr(?)dinger 方程(?) (1)H.Brézis 在文[1]中证明了,如果 Ω=R~2,p=3,K≤0或k>0,同时K integral from R~2|u_0|~2dx<4,(1)式有解 u(x,t),u∈C~1([0,∞),L~2(R~2))∩C([0,∞),H~2(R~2)).并且指出,在 Ω=R~2的情形,如果 p≥3,并且初值 u_0满足     

非线性梁振动方程的周期解

一、引言考虑下述问题Ku″ A~2u M(‖A~1/2u‖~2)Au Au′=f(x,t),t>0,x∈Ω,(1.1)u|_t=0~=u_0(x),x∈Ω,(1.2)Ku′|_(t=0)=u_1(x),x∈Ω,(1.3)u=0,x∈(?)Ω,t≥0 (1.4)的ω-周期解的存在性.其中 Ω(?)R~n 为一有界光滑区域,u′=((?)u)/((?)t),u_″=((?)u)/((?)t)~2,K 为有界线性对称算子且满足(Ku,u)≥0,M∈C~1[0,∞),M(ξ)≥-β,ξ≥0.此模型最初由Woinowsky 和 Krieger 提出,方程形式为     

Jacobian-determinant equation in the critical Besov spaces

Let Ω be a bounded domain in R~n with smooth boundary. Here we consider the following Jacobian-determinant equation det u(x)=f(x),x∈Ω;u(x)=x,x∈?Ω where f is a function on Ω with min_Ω f = δ 0 and Ωf(x)dx = |Ω|. We prove that if f ∈B_(p1)~(np)(Ω) for some p∈(n,∞), then there exists a solution u ∈ B_(p1)~(np+1)(Ω)C~1(Ω) to this equation. On the other hand, we give a simple example such that u ∈ C_0~1(R~2, R~2) while detu does not lie in B_(p1)~(2p)(R~2) for any p∞.     

一类非线性Schr dinger方程的高精度守恒数值格式

1引言本文讨论下面非线性Schr(?)dinger方程(NLS)方程的初边值问题:i(?)u/(?)t (?)~2u/(?)x~2 2|u~2|u=0,(1) u(x_l,t)=u(x_r,t)=0,t>0,(2) u(x,0)=u_0(x),x_l≤x≤x_r,(3)其中u(x,t)是复值函数,u_0(x)为已知的复值函数,i~2=-1.该问题有着如下的电荷与能量守恒关系:     

一类非线性Schrdinger-Maxwell方程基态解的存在性

本文研究了如下Schrdinger-Maxwell方程基态解的存在性问题{-△u+V(x)u+K(x)φ(x)u=b(x)|u|p-1u+λg(x,u)in R~3,-△φ=K(x)u~2in R~3,其中λ0,V(x)∈C~1(R~3,R),且V(x)0.△在K,g,b满足一定的假设条件下,且0p1时,利用变分法和临界点理论,获得了基态解的存在性.该结论推广了文献[7]的结果.     

一类广义Schrdinger型非线性高阶方程组

复函数的Schrdinger方程 u_1-iu_(xx)+β|u|~p u=0,p≥0 (1) 与复函数Schrdinger方程组 u_1-iu_(xx)+2u(a|u|~2+β|v|~2)=0 v_1-iv_(xx)+2v(a|u|~2+β|v|~2)=0 (2) 都可以看作一类实向量函数u=(u_1,u_2,…,u_j)的方程组 的特殊例子,其中A(t)是非奇异,非负定的J×J矩阵值函数,右边项向量函数f(u)的Jacobi矩阵f(u)/u是半有界的,这类方程组可称为广义Sehrdinger型方程组。     

带有Sobolev临界指标项的非齐次椭圆方程的解

本文考虑如下带有Sobolev临界指标项的非齐次椭圆方程{-?u=λu+|u|~(2*)-~2u+f,x∈?,u=0,x∈??,这里2~*=2N/N-2是Sobolev临界指标,N≥3,??R~N是一个有界开区域.0≤λλ_1,这里λ_1是算子-?的第一个特征值,并且假设f∈H_0~1(?)~(-1),当f满足适当的条件时,此方程在H_0~1(?)中至少具有两个解u_0和u_1.而且,当f≥0时,u_0≥0和u_1≥0.     

一类非线性椭圆边值问题解的存在性

目前 ,对 s——拉普拉斯算子△s的研究是较为活跃的数学课题 .原因在于算子 -△s与许多物理现象有关 .比如 :反射扩散问题 ,石油提取问题等等 .基于此因 ,在文 [3]的基础上 ,我们将继续研究以下非线性边值问题在 Ls(Ω) ,( 1 2 nn+1 )中解的存在条件 .-△su +g( x,u) =f几乎处处在Ω中-〈 ,| u|s- 2 u〉 =0几乎处处在Γ上其中 f∈Ls( Ω)给定 ,Ω Rn( n 1 ) ,△su=div( | u|s- 2 u) ,g∶Ω× R→ R满足 Caratheodory条件 .本文把文 [3]关于非线性边值问题 @在 Lp( Ω) ( 2 p<+∞ )空间中解的存在性的研究推广到 Ls( Ω) ( 1 2 nn+1 )空间中 .     

Scattering Versus Blowup Beyond the Mass-Energy Threshold for the Davey–Stewartson Equation in R~3

We study the Cauchy problem for the Davey–Stewartson equation i?_tu + Δu + |u|~2 u + E_1(|u|~2)u = 0,(t, x) ∈ R × R~3.The dichotomy between scattering and finite time blow-up shall be proved for initial data with finite variance and with mass-energy M(u_0)E(u_0) above the ground state threshold M(Q)E(Q).     

GLOBAL WELL-POSEDNESS IN ENERGY SPACE OF SMALL AMPLITUDE SOLUTIONS FOR KLEIN-GORDON-ZAKHAROV EQUATION IN THREE SPACE DIMENSION

The Cauchy problem of the Klein-Gordon-Zakharov equation in three dimensional space {utt-?u + u =-nu,(x, t) ∈ R~3× R_+,ntt-?n= ?|u|~2,(x, t) ∈ R~3× R_+,u(x, 0) = u_0(x), ?_tu(x, 0) = u_1(x),n(x, 0) = n_0(x), ?_tn(x,0) =n_1(x),(0.1) is considered. It is shown that it is globally well-posed in energy space H~1× L~2× L~2× H~(-1) if small initial data(u_0(x), u_1(x), n_0(x), n_1(x)) ∈(H~1× L~2× L~2× H~(-1)). It answers an open problem: Is it globally well-posed in energy space H~1× L~2× L~2× H~(-1) for 3D Klein-GordonZakharov equation with small initial data [1, 2]? The method in this article combines the linear property of the equation( dispersive property) with nonlinear property of the equation(energy inequalities). We mainly extend the spaces F~s and N~s in one dimension [3] to higher dimension.     

GLOBAL EXISTENCE AND BLOW-UP OF SOLUTIONS FOR GENERALIZED POCHHAMMER-CHREE EQUATIONS

In this article,we study the initial boundary value problem of generalized Pochhammer-Chree equation u_(tt)-u_(xx)-u_(xxt)-u_(xxtt)=f(u) xx,x ∈Ω,t 0,u(x,0) = u0(x),u t(x,0)=u1(x),x ∈Ω,u(0,t) = u(1,t) = 0,t≥0,where Ω=(0,1).First,we obtain the existence of local W k,p solutions.Then,we prove that,if f(s) ∈ΩC k+1(R) is nondecreasing,f(0) = 0 and |f(u)|≤C1|u| u 0 f(s)ds+C2,u 0(x),u 1(x) ∈ΩW k,p(Ω) ∩ W 1,p 0(Ω),k ≥ 1,1 p ≤∞,then for any T 0 the problem admits a unique solution u(x,t) ∈ W 2,∞(0,T;W k,p(Ω) ∩ W 1,p 0(Ω)).Finally,the finite time blow-up of solutions and global W k,p solution of generalized IMBq equations are discussed.     

“关于厄米特矩阵的一个不等式”的几点注记

<正> 文[1]提出并证明了下面的定理.设 A_j,B_j,…,C_j(j=1,2,…,k) 都是正定的同阶 (≥2) 厄米特矩阵,α,β,…,γ都是正实数,且 α+β+…+γ=1,则有sum from i=1 to k|A_j|~α|B_j|~β…|C_j|~γ<|sum from i=1 to k A_i|~α·|sum from i=1 to k B_i|~β…|sum from i=1 to k C_i|~γ.以下几点意见,供参考.第一,文[1]中的引理1和引理2是早有的结果.引理1见[2]p.15,[3]p.16及p.13,引理2是 Minkowski 行列式定理的直接推论,见[4].事实上,文[1]的定理是 H(?)lder 不等式和 Minkowski 行列式定理的自然结果.因为     

THE GLOBAL SMOOTH SOLUTIONS OF SECOND ORDER QUASILINEAR HYPERBOLIC EQUATIONS WITH DISSIPATIVE BOUNDARY CONDITIONS

The paper deals with the following boundary problem of the second order quasilinear hyperbolic equation with a dissipative boundary condition on a part of the boundary:u_(tt)-sum from i,j=1 to n a_(ij)(Du)u_(x_ix_j)=0, in (0, ∞)×Ω,u|Γ_0=0,sum from i,j=1 to n, a_(ij)(Du)n_ju_x_i+b(Du)u_t|Γ_1=0,u|t=0=φ(x), u_t|t=0=ψ(x), in Ω, where Ω=Γ_0∪Γ_1, b(Du)≥b_0>0. Under some assumptions on the equation and domain, the author proves that there exists a global smooth solution for above problem with small data.     

磁微极流体方程组在临界Sobolev空间强解的存在性

在临界Sobolev空间H~(1/2)(R~3)中,本文研究了三维不可压磁微极流体方程组的适定性.设(u_0,ω_0,b_0)是H~(1/2)(R~3)中的小初值,则三维不可压磁微极流体方程组存在唯一整体强解(u,ω,b)∈C([0,+∞);H~(1/2)(R~3))∩L~2((0,+∞);H~(3/2)(R~3))∩L~4((0,+∞);H~1(R~3));设大初值(u_0,ω_0,b_0)∈H~(1/2)(R~3),则存在一个正的时间T=T(u_0,ω_0,b_0)使得三维不可压磁微极流体方程组在[0,T]内存在唯一局部强解(u,ω,b)∈C([0,T];H~(1/2)(R~3))∩L~2((0,T];H~(3/2)(R~3))∩L~4((0,T];H~1(R~3)),这些改进了Yuan J的结果(Existence theorem and blow-up criterion of the strong solutions to the magnetomicropolar fluid equations,Math.Methods Appl.Sci.,31(2008),1113-1130).     

常微分方程边值问题分歧解的例子

本文的目的是给出简单非线性二阶常微分方程边值问题具有分歧解的一些例子。在区间[-R,R]上,考虑常微分方程及边值条件 u(-R)=u_0,u(R)=u_0, (2)其中R>0,u_0>0为给定常数,λ为参数,已知函数K(u)与F(u)是光滑的,并且当u>0时K(u)>0,F(u)>0。从方程及边值条件的对称性,可知当x=0时u＇(0)=0。记u(0)=u~*为待定常数。积分一次得     

GLOBAL EXISTENCE OF THE SOLUTIONS TO NONLINEAR HYPERBOLIC EQUATIONS IN EXTERIOR DOMAINS

This paper deals with the following IBV problem of nonlinear hyperbolic equations u_(tt)- sum from i, j=1 to n a_(jj)(u, Du)u_(x_ix_j)=b(u, Du), t>0, x∈Ω, u(O, x) =u~0(x), u_t(O, x) =u~1(v), x∈Ω, u(t, x)=O t>O, x∈()Ω,where Ωis the exterior domain of a compact set in R~n, and |a_(ij)(y)-δ_(ij)|= O(|y|~k), |b(y)|=O(|y|~(k+1)), near y=O. It is proved that under suitable assumptions on the smoothness,compatibility conditions and the shape of Ω, the above problem has a unique global smoothsolution for small initial data, in the case that k=1 add n≥7 or that k=2 and n≥4.Moreover, the solution ham some decay properties as t→ + ∞.     

Self-maps of p-local infinite projective spaces

Denote by Z (p) (resp.Z p ) the p localization (resp.p completion) of Z.Then we have the canonical inclusion Z (p) → Z p .Let S 2n-1 (p) be the p-local (2n-1)-sphere and let B 2n (p) be a connected p-local space satisfying S 2n-1 (p) ～= ΩB 2n (p) ;then H - (B 2n (p) ,Z (p) ) = Z (p) [u] with |u| = 2n.Define the degree of a self-map f of B 2n (p) to be k ∈ Z (p) such that f *(u) = ku.Using the theory of integer-valued polynomials we show that there exists a self-map of B 2n (p) of degree k if and only if k is an n-th power in Z p .     

一阶微分方程的渐近平稳和周期边值问题

假定函数 f∈C[R_+×R,R],我们考虑非线性问题u'=f(t,u),u(t_0)=u_0,t_0≥0.(A)[1]附录的定理 A.1.2就(A)的渐近平稳(Asymptotic Equilibrium)给出如下的定理 A。假定 g(t,u)∈C[R_+×R_+,|R_+]对于每个 t 关于 u 单调非减,且使得|f(t,u)|≤g(t,|u|),(t,u)∈R_+×R.如果问题u′=g(t,u),u(t_0)=u_0≥0的所有解 u(t)在[t_0,∞)上有界,那么问题(A)渐近平稳.利用这个定理,[1]在假定,f(t,u)满足单边的 Lipschitz 条件     

BRILL－NOETHER MATRIX FOR RANK TWO VECTOR BUNDLES

Lei X be an arbitrary smooth irreducible complex projective curve, E (?) X a rank two vector bundle generated by its sections. The author first represents E as a triple {D1,D2,f}, where D1 , D2 are two effective divisors with d = deg(D1) + deg(D2), and f ∈ H0(X, [D1] |D2) is a collection of polynomials. E is the extension of [D2] by [D1] which is determined by f. By using f and the Brill-Noether matrix of D1 + D2, the author constructs a 2g X d matrix WE whose zero space gives Im{H0(X,[D1]) (?) H0(X, [D1] |D1)}(?)Im{H0(X, E) (?) H0(X,[D2]) (?) H0(X,[D2] |D2)}. From this and H0(X,E) = H0(X, [D1]) (?) Im{H0(X, E) (?) H0(X, [D2])}, it is got in particular that dimH0(X, E) = deg(E) - rank(WE) + 2.     

一类具有奇异性与真空的非牛顿流局部强解的爆破准则

探讨了如下一类非牛顿流pt+(pu)χ=0,(pu)χ+(pu2)χ-(︱uχ︱p-2uχ)χ+πχ=pf,π=π(p)=Apr,(χ,t)∈Ωr1,A>0,r>1,其初边值条件为(p,u)|t=0=(p0,u0),χ∈(-1,I),u|χ=1=u|χ=-1=0,t∈(0,T1).利用迭代方法,讨论了该模型的局部强解的爆破准则,证明了:如果T_*是强解(ρ,u)存在的最大时间且T*相似文献